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Solving Inequalities By: Sam Milkey and Noah Bakunowicz

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Polynomial Inequalities A polynomial inequality takes the form f(x) > 0, f(x) ≥ 0, f(x) < 0, f(x) ≤ 0, or f(x) ≠ 0. To solve f(x) > 0 is to find the values of x that make f(x) positive. To solve f(x) < 0 is to find the values of x that make f(x) negative.

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But that’s pretty boring. https://www.youtube.com/watch?v=_J7xwaOrnf8 (skip to 1:00)

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Example 1 Finding negative, positive, zero F(x)=(x+2)(x+1)(x-5) Zeros: -2 (mult of 1), -1(mult of 1), 5 (mult of 1) Number line: Find when it is Zero, Negative, and Positive o Zeros: -2, -1, 5 o Negative: (∞, -2) (-1, 5) o Positive: (-2,-1) (5,∞)

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Example 2 Solving Algebraically Solve 2x³-7x²-10x+24>0 Analytically Use the rational zeros theorem to find possible rational zeros o ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, ±1/2, ±3/2 You can use a graph to figure out which zero to use first, in this case x=4 is good.

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Example 2 cont. Using synthetic division Factor 2x²+x-6 o (2x-3)(x+2) So f(x)=(x-4)(2x-3)(x+2) Zeros= 4, 3/2, F(x)=(x-4)(2x²+x-6 )

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Example 2 cont. Sign Chart You can find the where it is negative or positive from its end behavior Since we wanted to find out when it is greater than 0, the solutions are (-2,3/2) and (4,∞) -23/2 4 Sign Change -+-+

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Example 3 Solving Graphically Solve x 3 -6x 2 ≤ 2-8x graphically Rewrite the inequality so it is less than or equal to 0 o x 3 -6x 2 +8x-2 ≤ 0 Type in x 3 -6x 2 +8x-2 into the y1 of the graph on your calculator o Zeros are approximately 0.32, 1.46, and 4.21 Since we want when it is less than 0, we want all of the numbers below the x-axis on the graph o Solution: ( -∞,0.32] and [1.46, 4.21] Remember, use hard brackets because those points are solutions too!

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Example 4 Solving with Unusual Answers The inequalities associated with a strictly positive polynomial function such as f(x) = (x 2 +7)(2x 2 +1) have strange solutions o (x 2 +7)(2x 2 +1) > 0 is all real numbers o (x 2 +7)(2x 2 +1) ≥ 0 is all real numbers o (x 2 +7)(2x 2 +1) < is no solution o (x 2 +7)(2x 2 +1) ≤ is no solution

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Example 4 Cont. The inequalities associated with a nonnegative polynomil function such as f(x)=(x 2 -3x+3)(2x+5) 2 also has strange answers o (x 2 -3x+3)(2x+5) > 0 is (-∞,-5/2) and (-5/2,∞) o (x 2 -3x+3)(2x+5) ≥ 0 is all real numbers o (x 2 -3x+3)(2x+5) < 0 has no solution o (x 2 -3x+3)(2x+5) ≤ 0 is a single number, -5/2

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Example 5 Creating Sign Charts Let f(x) = (2x+1)/((x+3)(x-1)). Find when the function is (a) zero (b) undefined. Then make a sign chart to find when it is positive or negative. (a). Real zeros of the function are the real zeros of the numerator. in this case 2 x+1 is the numerator (b). f(x) is undefined when the denominator is 0. Since (x+3)(x-1) is the denominator, it is undefined at x = -3 or x = 1. Sign Chart -3-1/21 Potential Sign Change

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Example 5 cont. Sign chart with undefined, zeros, positive, and negative f(x) is negative if x < 3 or -1/2 < x < 1, so the solutions are (-∞, -3) and (- 1/2, 1) f(x) is positive if -3 1, so the solutions are (-3, -1/2) and (1,∞) -3-1/21 (-) (-)(-) und. 0 (+)(-) (-) (+)(-) (+) (+)(+) - +-+

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Example 6 Solve by Combining Fractions Solve (5/(x+3))+(3/(x-1)) < 0 5 x x-1 < 0 Original Inequality (x+3)(x-1) 5(x-1) + (x+3)(x-1) 3(x+3) < 0 Use LCD to rewrite fractions (x+3)(x-1) 5(x-1) + 3(x+3) < 0 Add Fractions

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Example 6 cont. 5x-5+3x+9 (x+3)(x-1) < 0 Distributive property (x+3)(x-1) 8x+4 < 0 Simplify (x+3)(x-1) 2x+1 < 0 Divide both sides by 4 Solution: (-∞, -3) and (-1/2, 1).

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Example 7 Inequalities Involving Radicals Solve (x-3)√(x+1) ≥ 0. Because of the factor √(x+1), f(x) is undefined if x < -1. The zeros of f are 3 and -1. Sign Chart: Solution: {-1} and [3, ∞) 0 0 UndefinedNegativePositive (-)(+)(+)(+) 3

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Example 8 Inequalities with Absolute Value Solve x-2 Because x+3 is in the denominator, f(x) is undefined if x = -3. The only zeros of f is 2. Solution: (-∞, -3) and (-3,2] x+3 ≤ 0 Negative Positive (-) - + (+) +

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Matching Game The link for the game can be found here Grading Scale A = 60 seconds or less B = in between 60.1 and 90 seconds C = in between 90.1 and 120 seconds D = in between and 150 seconds F = Anything greater than seconds

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Quiz 1.) Combine the fraction and reduce your answer to lowest terms. x 2 +5/x A.) (x + 5)/x 3 B.) (x 3 + 5)/x C.) (x + 5) 3 /x 2.) Which one of these is a possible rational zero of the polynomial. 2x 3 +x 2 -4x-3 A.) ±4 B.) ±2 C.) ±3 D.) All the above 3.) Determine the x values that cause the polynomial function to be a zero. f(x) = (2x 2 +5)(x-8) 2 (x+1) 3 A.) 8 B.) -1 C.) 5 D.) A and B E.) All the above

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Quiz Page 2 4.) The graph of f(x) = x 4 (x+3) 2 (x-1) 3 changes sign at x = 0. A.) True B.) False 5.) Which is a solution to x 2 < x A.) (1, ∞) B.) (0,1) C.) (0, ∞) 6.) Solve the inequality. x|x - 2| > 0 A.) (0,2)U(2,∞) B.) (-∞, 2)U(2,∞) C.) None of these answers 7.) Solve the polynomial inequality. x 3 - x 2 - 2x ≥ 0 A.) [-2,0]U[1,∞) B.) [-1,0]U[2,∞) C.) [0,1]U[2,∞)

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Quiz Page 3 8.) Complete the factoring if needed and solve the polynomial inequality. (x + 1)(x 2 - 3x + 2) < 0 A.) [-1,0]U[2,∞) B.) (-∞,0)U(2,3) C.) (-∞,-1)U(1,2) 9.) Dunder Mifflin Paper Company wishes to design paper boxes with a volume of not more than 100 in 3. Squares are to be cut from the corners of a 12-in. by 15-in. piece of cardboard, with the flaps folded up to make an open box. What size squares should be cut from the cardboard. A.) 0 in. ≤ x ≤ 0.69 in. B.) 0 in. ≥ x ≥ 0.69 in. C.) 4.20 ≤ x ≤ 6 in. D.) 4.20 ≥ x ≥ 6 in. E.) A and C F.) B and D 10.) Solve the polynomial inequality. 2x 3 - 5x 2 - x + 6 > 0 A.) (-1, 3/2)U(2,∞) B.) [-1, 3/2]U[2,∞] C.) (-1, 3/2]U[2,∞)

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Answer Key 1.) B 2.) C 3.) D 4.) False 5.) B 6.) A 7.) B 8.) C 9.) E 10.)

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Work Cited Precalculus Graphical, Numerical, Algebraic; Eighth Edition https://www.youtube.com/watch?v=_J7xwaOrnf8 (malakai333) culator/graphCalc.html

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